How do I stop a displacement modifier from breaking the edges of its mesh into many small parts?
With the displacement modifier, the edges are occluded by bumps created by the modifier and hence are no longer fully visible from the camera.  Freestyle identifies both visible and hidden lines, and by default draws only visible lines.  If you want to draw all lines no matter whether they are visible or hidden, then you need to turn off the selection by visibility (i.e., disable the "Visibility" toggle in the "Selection By" section of the Freestyle Line Set panel).

Update on May 12, 2017

Setting the mesh faces to the smooth shading and turning on the Face Smoothness option could improve the result as you expect.

That said, there seems a bug in Freestyle that prevents visible and hidden lines from getting chained into a single stroke.  For instance, the following render shows lines composed of visible and hidden parts.  Most lines are consistent in the sense that they start in red and end in green, while some lines are not.



The next image shows some of those inconsistent lines that are expected to be long continuous red-green strokes rather than being split into short pieces.



The render below shows the visibility of line segments in question (visible and hidden parts in red and green, respectively).



By comparing the last two renders, it is seen that in some cases visible and hidden line segments can be chained to form a consistent line, while not in some other cases.  I think this is a bug in Freestyle (to be fixed as soon as time permits).